What is parallax error

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Tuesday, December 22, 2009

what is a parallax error?

Parallax error is the error that is most committed when readings are taken in physics. You can thus understand why it is important to avoid it at all cost. One must be aware of its existence at all time so that it can be avoided and as a result the true value of the reading is obtained.

The concept of parallax error is related to the term parallax.

Imagine that we have in a room a pelican and a flamingo as shown in fig 1 below.

Now Garfield is moving about in the room and each time is is somewhere in the room, he looks at the two birds. At the point A he sees the flamingo on the left of the pelican whereas when he is at position C he would see the flamingo on the right of the pelican. Only when he would be at position B would he sees the two birds one behind the other.

He would discover that each time he is at a different position, he would find that the position of the flamingo relative to the pelican has changed.

You can also have this effect when you are in front of a clock. If you move from side to side you would find that the time that you can read from the clock is different.

So we can the understand that the parallax is the change in the apparent position of an object when the position of the observer changes.

Now let us look at the concept of parallax error. If you have placed a pencil on a metre rule and you are reading its length then just like in fig 1 above and in fig 2 below you can place you eye everywhere you want.

As you can see from fig 2 above I have chosen three position at which you can place your eye.

Clearly at these three position we can have the following reading

Reading at A = 6.2 cm

Reading at B = 5.8 cm

Reading at C = 5.5 cm

What you you think would be the correct reading?

The correct reading is would be obtained when the eye is placed at B.

Now there is a line from the tip of eye to the of the pencil that continues up to the scale. this line is called the line of sight and the mark at which the line intersect the scale is the length of the pencil. This line of sight must be be at right angle to the scale. This is shown above in fig

If the line of sight and the scale are not ar right angle to each other then a parallax error is committed.

Similarly with a measuring cylinder the line of sight from the eye to the bottom of the meniscus must be at right angle to the scale as shown in fig 4 below. In this case the line of sight is horizontal and the scale vertical.

So as you can see above each time you are taking a reading you must ensure that the line of sight is perpendicular to the scale.

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Parallax is great! Gaia will expand our view

What is parallax?

Parallax is the apparent shift in position of a nearby object when it is viewed from different locations. Those might be two different observatories on the Earth’s surface (as was the case in the famous transits of Venus in the eighteenth and nineteenth centuries), but, in most astronomical applications, are two spots in Earth’s orbit around the Sun:

If one knows the value of the baseline distance b, and can measure the parallax angle π

Yes, I know this is confusing. Yes, astronomers back in the old days should have picked a better letter. Sometimes you see the «curly pi» used to denote this angle, but few font families include it

then one can use simple geometry to compute the distance to the target object d.

Of course, that simple formula ignores many complications. For example,

• the target object is usually located far above or below the plane of the Earth’s orbit
• the target star and the Sun are both moving through the Milky Way Galaxy
• the background reference stars are not located at an infinite distance, and so suffer their own small parallactic shifts
• the background stars aren’t even all at the same distance

But one can work around these issues, or at least place constraints on their consequences.

Why is parallax such a good method?

The importance of parallax lies in its fundamental simplicity: we humans understand geometry, and trust it. As long as we can make measurements of the angular displacements over time, we can use the resulting distances with confidence.

There may be minor issues involving corrections for the small shifts of the background reference stars themselves; but if we choose to use galaxies or quasars as the reference sources, we can avoid those completely.

Most important of all, parallax is a direct method of measuring distances. It relies on a single assumed quantity: the astronomical unit (AU) = the distance from Earth to Sun. Back in the 1800s and early 1900s, this was indeed a major issue, and astronomers great efforts to pin down the value of the AU. But these days, thanks to radar and spacecraft flying through the solar system, we do have a very, very good idea for the size of the AU. For example, the JPL Horizons system uses a value taken from the Planetary and Lunar Ephemerides DE430 and DE431, which list

That’s . quite a few signficant figures.

And that’s it. Heliocentric parallax does NOT require any assumptions about stellar luminosity or evolution, or the distribution of sizes and colors of stars, or the spectral energy distribution of glowing hydrogen gas.

So, if astronomers can measure a distance via parallax, they will rely on it much more than distances determined by other methods.

How far can it reach? (Classical)

So, parallax is a good method. But how far out into space can it reach?

Back in the old days, when parallax was performed by optical telescopes from the surface of the Earth, it was very difficult to achieve precisions of better than about 0.01 arcsec. Let’s use milliarcseconds (mas) from this point forward, as it will be more convenient. So, good ground-based measurements had precisions of perhaps 10 mas.

The very best, made with specially-built instruments such as the Multichannel Astrometric Photometer at Allegheny Observatory , could achieve precision of 1 mas for a few stars, after many observations.

However, no such devices could make all-sky surveys. As a result, catalogs of stellar distances were incomplete, as well as being rather low in precision. For example, the Bright Star Catalog , compiled over the period of 1964 to 1991, listed parallax values for about 2700 stars.

The fact that this histogram turns over below 15 mas is a sign that the precision of the typical measurement is . about 15 mas.

The annoying nature of errors in parallax measurements: part I

It turns out that the nature of parallax measurements means that the connection between the error in the measurement and the error in the distance is not simple. There are a number of tricky aspects which can make the analysis of, say, the possible bias in some catalog, a complicated matter.

The first big problem is that the quantity of interest, the distance to a star d, is INVERSELY related to the measured angle π.

This means that an error of (for example) 20 percent in the measurement does NOT mean that there will be an error of 20 percent in the computed distance.

Why not? Well, let’s do an example to find out. Suppose that we measure the parallax to a star to be

If the uncertainty is the usual «1-sigma» variety, then this measurement means that the true value of the parallax angle has a 66% chance of lying in the range 80 to 120 mas.

Fine so far. But what does that mean for distance we derive from this angle? We need to calculate the distance corresponding to the angle (100 — 20) and the angle (100 + 20). I’ll let you fill in the table below.

The fundamental issue here is that a realistic, symmetric error in the measured angle leads to an ASYMMETRIC error in the derived distance. The larger the fractional error in parallax angle becomes, the more asymmetric this error in the derived distance. For example, if we measure

then the range of distances consistent with the measurement at 1-sigma looks like this:

We can sum up this first problem: symmetric errors in a measured parallax angle lead to asymmetric errors in the derived distance . with a larger range of possible true distances on the «farther away» side.

The annoying nature of errors in parallax measurements: part II

The really nasty bit of the errors-in-parallax discussion is a consequence of the first conclusion: symmetric errors in angle lead to asymmetric errors in distance. Let’s go back to this example again:

Based on the parallax angle π = 10 mas, we derive a distance of d = 100 pc — but due to the uncertainty in the angle, the true location of the star has a 66% chance of lying anywhere within the range of 59 — 333 pc.

What that means is that the TRUE location of the star could be here .

In fact, the star could lie anywhere along a line in this direction, with equal probability, based on that one measurement. In a one-dimensional universe, the probability that the TRUE location of the star is farther than 100 pc is larger than the probability that the star is closer than 100 pc, simply by the ratio of the lengths of the possible locations:

In a TWO-DIMENSIONAL universe, the probabilities would change because the region in which a «farther-than-100-pc» star could exist becomes much larger than that of a «closer-than-100-pc» star.

Our course, our real universe is THREE-DIMENSIONAL. Sorry, I can’t draw a good 3-D figure showing the two regions, but I hope you can extrapolate from the two-dimensional case to the three-dimensional one.

Now, what’s the big deal about all of this? The big deal is that we live in a region of relatively uniform stellar density. In our neighborhood of the Milky Way, there are roughly equal numbers of stars in all directions: left, right, up, down, forward, back. Yes, there tend to be more stars in the plane of the Milky Way, but the thickness of the disk is (until Gaia) larger than typical distance we could measure via parallax, that doesn’t really matter.

And so, suppose that we measure the parallax to a star to have the value 10 +/- 1 mas, corresponding to a measured distance of 100 pc. But there are three possibilities for the TRUE distance to this star:

the star could actually be closer to the Earth, but a negative error in the measured parallax (taking a true value of 11 mas to measured value of 10 mas) makes it appear to be farther away

the star could actually be at exactly 100 pc

Which of these is most likely? The answer is «C». Because there are more stars in the «more distant» volume, it is more likely that a distant star has been improperly measured than a nearby star. The result is that our measurements of parallax lead to a systematic bias: stars appear closer than they really are (sort of like dinosaurs.)

This effect was noted long ago, by Trumpler and Weaver (1953), among others, but Lutz and Kelker first analyzed it detail in 1973. Astronomers sometimes refer to this bias, or the corrections needed to account for it, as the «Lutz-Kelker effect.»

The size of this bias depends (mostly) on the ratio of the uncertainty in parallax to the value of parallax:

Let’s look at some examples. We will observe a set of stars with measured parallax angles of exactly 10.0 mas, which means measured distances of exactly 100 pc.

When the fractional error is very small — just a few percent — then the errors in the derived distances, and the overall bias, are pretty small.

But as the fractional uncertainty grows, the errors and bias increase. Even uncertainties of just 10% can cause big mistakes in the measured parallax. Notice the long tail of true distances out to 140 pc and beyond; the errors in those derived distances are 40 percent or more, far larger than the 10% uncertainty in the parallax angle itself.

If the uncertainty becomes 20% or larger, the results involve errors which are frequently a factor of 2 or more. No one should try to use such measurements for any careful calculations.

Things are just silly when the uncertaity is 30%!

This table summarizes the results of a simulation in which I created a region of space with uniform stellar density, «measured» enough stars to find 1000 falling within a particular range of parallax values around 10 mas, and then compared those «measured» distances to the actual distances.

The moral of this story is — don’t trust distances based on parallax unless the uncertainty in the measurement is much smaller than the measurement itself. I would prefer not to rely on any measurements in which the fractional uncertainty is larger than 10 percent.

How far can it reach? (Classical, again)

Phew. Now that we understand the weaknesses of the parallax method, we can try to answer this question. If we adopt as a general rule that most ground-based measurements of parallax had uncertainties of about σ_π = 10 mas, then we can place a very rough limit on the distances of stars derived from such measurements:

Gosh, that’s not very far.

Of course, astronomers DID use parallax measurements which were considerably smaller than 100 mas; scientists are always interested in pushing the limits of their data. But this is a warning that any statistical inferences based on ground-based measurements of stars at greater distances must be very carefully considered, and corrections must be made for the Lutz-Kelker effect.

How far can it reach? (Hipparcos)

The Hipparcos satellite was launched in 1989 and its first catalog was released in 1997. This project immediately improved our knowledge of stellar distances (and motions, and luminosities) by a large amount. Not only did it cover the entire sky, and measure stars fainter than those typically found in parallax catalogs (down to about mag 10), it also had a typical precision of about 2 mas. In other words, its typical measurement was about as good as the very best ground-based measurements.

Image of M101 courtesy of NASA, ESA, K. Kuntz (JHU), F. Bresolin (University of Hawaii), J. Trauger (Jet Propulsion Lab), J. Mould (NOAO), Y.-H. Chu (University of Illinois, Urbana), and STScI; Canada-France-Hawaii Telescope/ J.-C. Cuillandre/Coelum; G. Jacoby, B. Bohannan, M. Hanna/ NOAO/AURA/NSF

One of the wonderful results of Hippacos’ measurements were the very accurate HR diagrams which could finally be made, since we now had LOTS of distance measurements to stars of all types. Below is one made using Hipparcos stars with uncertainties of ≤ 10% in their parallax angles.

Image courtesy of ESA’s page with HR diagrams based on Hipparcos data.

How far can it reach? (Gaia)

Gaia is a rather unusual «telescope,» since it has been specially designed for astrometric observations. It resembles the Hipparcos satellite in many ways. The basic structure is built around two triple-mirror telescopes which point at regions in the sky separated by an angle of about 106.5 degrees.

Image courtesy of Wikipedia

Light from the two telescopes is combined so that it falls upon the focal plane simultaneously. The telescope rotates with a period of six hours, causing stars to sweep across this focal plane in about 60 seconds. Objects in both directions are detected and measured together.

Image courtesy of Wikipedia

Figure 1 from Lindegren et al. 2016. Note that this figure is rotated by 180 degrees relative to previous schematic. The original caption follows.

Layout of the CCDs in Gaia’s focal plane. Star images move from left to right in the diagram. As the images enter the field of view, they are detected by the sky mapper (SM) CCDs and astrometrically observed by the 62 CCDs in the astrometric field (AF). Basic-angle variations are interferometrically measured using the basic angle monitor (BAM) CCD in row 1 (bottom row in figure). The BAM CCD in row 2 is available for redundancy. Other CCDs are used for the red and blue photometers (BP, RP), radial velocity spectrometer (RVS), and wave- front sensors (WFS). The orientation of the field angles η (along-scan, AL) and ζ (across-scan, AC) is shown at bottom right. The actual origin (η, ζ) = (0, 0) is indicated by the numbered yellow circles 1 (for the preceding field of view) and 2 (for the following field of view).

The telescope slowly precesses, so that its spin sweeps out new areas in the sky over time. A typical star will be measured about 70 times over the course of the primary mission (2014 — 2019); that’s about 14 measurements per year, though not all equally spaced.

Image courtesy of Space Flight 101

As the spacecraft precesses, the «partners» of each star will gradually change. Eventually, the spacecraft will have many millions (billions?) of measurements of the relative positions of millions of stars. Scientists can then use a honking-big linear algebra procedure to solve simultaneously for the positions and motions of each star.

How precise are the measurements made by Gaia? As described in Lindegren et al. 2016, the Gaia Data Release 1 consists of a limited set of measured properties for a very limited set of bright stars. Later releases will contain (much) better data for (many) more stars.

Image of M101 courtesy of NASA, ESA, K. Kuntz (JHU), F. Bresolin (University of Hawaii), J. Trauger (Jet Propulsion Lab), J. Mould (NOAO), Y.-H. Chu (University of Illinois, Urbana), and STScI; Canada-France-Hawaii Telescope/ J.-C. Cuillandre/Coelum; G. Jacoby, B. Bohannan, M. Hanna/ NOAO/AURA/NSF

Gaia provides astronomers with many, many more stars than Hipparcos did, so it is possible to create much more detailed HR diagrams:

As time passes, and Gaia continues to scan the skies, it will build up larger and larger sets of measurements of each star, allowing it to create a better solution for the position, parallax, and proper motion of each object. A recent paper by de Bruijne, Rygl and Antoja provides some predictions for the FINAL Gaia precisions.

Fig 3 taken from de Bruijne, Rygl and Antoja (2015)

Image of M101 courtesy of NASA, ESA, K. Kuntz (JHU), F. Bresolin (University of Hawaii), J. Trauger (Jet Propulsion Lab), J. Mould (NOAO), Y.-H. Chu (University of Illinois, Urbana), and STScI; Canada-France-Hawaii Telescope/ J.-C. Cuillandre/Coelum; G. Jacoby, B. Bohannan, M. Hanna/ NOAO/AURA/NSF

In any case, Gaia will greatly increase our knowledge of the stellar neighborhood. It will produce a catalog of roughly 100 million stars with distances good to about 10 percent. The Hipparcos catalog contains only 0.12 million entries, of which only about 0.04 million satisfy the Lutz-Kelker criterion.

Let’s look at the timeline for Gaia operations and data releases.

Figure 8 taken from Eyer et al., 2015

Hipparcos vs. Gaia: The case of RR Lyr

As we will see in a future lecture, the star RR Lyrae is a very important one. It is one of the nearest and brightest of a class of pulsating stars which can help us to measure distances not only within the Milky Way, but to other galaxies.

Clearly, if we wish to use the class of RR Lyr stars as distance indicators, we need to know very precisely how luminous they are . and, so, how far away they are. Let’s consider the star RR Lyr itself.

Use the Vizier interface to the Gaia ‘TGS’ catalog to find the distance to RR Lyr

Do the two distances agree within their 1-sigma uncertainties?

This is, of course, just one star. But if our estimates of the distances to other stars, and other galaxies, depend upon RR Lyr stars in particular, and if the new Gaia data suggests that our old value might be incorrect by such an amount . well, that should make you start to worry about the upper rungs on the Cosmological Disetance Ladder!

Hipparcos vs. Gaia: The general trend

Based on the earlier discussion of errors in parallax , we might expect that the Hipparcos measurements, with their larger uncertainties, would show a systematic bias relative to the more precise Gaia values. Let’s put that to the test.

I went to the Gaia DR1 archive site and chose the «ADQL Search Form», which allows the user to enter SQL-like queries to a number of catalogs.

Into the query box, I entered

The result was a set of about 93,000 stars measured by both Hipparcos and Gaia. I then compared the parallax values on a star-by-star basis. As the graph below shows, Gaia and Hipparcos do agree very well, on average: the red dots are individual stars, and the blue symbols are medians within bins of width 1 milliarcsec.

If we zoom in, however, we can see some asymmetry: the Gaia parallax angles tend to be SMALLER than the Hipparcos angles, and the difference grows as the angles decrease in size.

In other words, since the Hipparcos angles are slight OVER-estimates (due to the Lutz-Kelker bias), the Hipparcos distance values tend to be UNDER-estimated as on approaches the limit of its measurements.

We can see that clearly if we plot DISTANCES on our graph instead of angles.

What about radio measurements?

Of course, optical telescopes are not the only ones which can make parallax measurements. Any telescope on (or around) the Earth will share the annual journal, and so have a chance to measure the parallactic shifts of celestial sources.

Stars emit most of their energy in the optical region of the spectrum, and very little in the radio; it isn’t possible to detect most stars with radio telescopes. Clouds of gas and dust do emit plenty of radio waves, but they are for the most part large, extended sources, not the compact objects one requires for parallax measurements.

However, there are some circumstances which do allow radio astronomers to apply the power of interferometry to make very precise measurements of the positions of compact radio sources, and therefore to determine the distances to those sources via parallax. The key element is the maser emission from small clumps of gas within some star-forming regions. Astronomical masers arise within compact volumes, thus appearing as point sources to our radio telescopes, and produce radiation of very pure frequency. The combination allows astronomers to determine their positions with very small uncertainties.

One example of parallax measurements in the radio regime is provided by the Japanese VERA project. VERA is a array of four radio dishes stretched out across the Japanese archipelago.

Image courtesy of The VERA group and the National Astronomical Observatory of Japan

If one looks at star-forming regions, such as S269, with an optical telescope, one sees clumps of stars immersed in gas and dust. If one looks with a radio interferometer, one can see much finer details. Compare the scales on the IR picture at left with the VERA map at lower right.

Image taken from a presentation by Mareki Honma, NAOJ

If one measures the position of one of these masers over a period of several years, one will see an intriguing pattern. Shown below are VERA measurements of a maser in the Orion-KL region; the figure shows the Right Ascension component of the position as a function of time.

Figure taken from a presentation by Mareki Honma, NAOJ ; I’ve erased the parallax result so that my students can’t peek at it in class.

A slightly more recent measurement by VERA yields the even more precise parallax to Orion-KL of

Now, if VERA can measure parallax with a precision of 0.03 mas, then it can reach far out into the Milky Way, too. In fact, at the moment, VERA is one of our best tools for measuring distances in the Milky Way, beating Gaia quite handily.

Image of M101 courtesy of NASA, ESA, K. Kuntz (JHU), F. Bresolin (University of Hawaii), J. Trauger (Jet Propulsion Lab), J. Mould (NOAO), Y.-H. Chu (University of Illinois, Urbana), and STScI; Canada-France-Hawaii Telescope/ J.-C. Cuillandre/Coelum; G. Jacoby, B. Bohannan, M. Hanna/ NOAO/AURA/NSF

For more information

• One of the lectures in my course on extragalactic astronomy and cosmology discuss parallax:
  • The AU, parallax, and other geometric methods

  Way back in the late 1990s, I discussed parallax and the (at that time) exciting new results from Hipparcos. You might find some of the illustrations of the «raw-ish» data from Hipparcos useful.

• Of course, you will want to get your hands on the real Gaia measurements. You have several choices:
  • Vizier catalog I/337 provides interactive searching and plotting access to Gaia’s TGAS catalog. I find TGAS more useful than the other Gaia DR1 products, since it includes parallaxes and proper motions.
  • The Gaia DR1 release page presents information about this data release, with pictures and statistics and links to many resources.
  • The Gaia archive web page has links to the actual catalog interfaces.
• I gathered some material on VERA for a public talk — perhaps it might be useful for reference:
  • A talk from 2008 featuring some results from VERA

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.

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The parallax error is one that occurs when a measurement is made from different viewing angles. A simple example of what is meant by parallax is seen when we close the left eye in front of an object, and then immediately open the left eye and close the right eye: the object will appear to have moved to the right. But when we have both eyes open, we will see the object in the middle.

The same phenomenon can be repeated by positioning our eyes at different angles to an object, such as a simple point (image below). If we wanted to place this point on a ruler, we would make a parallax error when viewing it from the positions marked by the X’s in red.

From the X on the right the point will appear to be further to the left, so its location with respect to the ruler is closer to 3 than to 4. Meanwhile, the opposite occurs from the X on the left: now the point will appear to be further to the right, placing it closer to 5 on the ruler.

To avoid parallax error in such measurements, it is always necessary to place the eyes in the most vertical position possible with respect to the scale of the instrument or material.

In the image above we had a scale, the point being a needle, a pointer, the edge of an object, the meniscus of a liquid, etc. That is, we have a scale on which we want to make a reading. Very similar as it happens with the hands of the clock when we read what time it is.

In a laboratory we have scales that offer hundreds of instruments or materials made of glass or metal. When a student makes a measurement that includes direct scaling, he is subject to making parallax errors, which is random: he constantly varies the angle from which he observes or takes the measurement.

Sometimes because of the rush and wanting to write down a reading, the student does not realize that he is making the parallax error, since it does not only occur from left or right angles; also from above or below, and there may be even other angles. You should always take the time to fix your eyes in the most upright and frontal position when comparing an object against a scale.

That is to say, in the laboratory the eyes must be as far as possible in front of the object of our measurements. Otherwise, the accuracy will suffer.

In chemistry laboratories we have in particular the parallax error in volume measurements. For this, there are glass materials such as the graduated cylinder or the burette, each with its own scales (think of a ruler but cylindrical and made of glass).

Depending on where and from what direction we want to see the meniscus, its apparent position on the scale will be different. Viewed from below, the meniscus will be lower (11.69 mL) than it is. But viewed from above the meniscus will now appear higher on the scale (11.76 mL).

The real volume (11.73 mL for our example) will be the one that we measure with the eyes (both) in a position as frontal and vertical as possible with respect to the glass material. As you will see in the examples section, there are many operations that generate parallax errors in a chemistry laboratory.

Parallax error in physics

Laboratories

In physics laboratories, on the other hand, volumes can also be measured, but other instruments are of greater importance in their practices.

For example, it is necessary to determine the dimensions, such as lengths and width of an object, using the micrometer (palmer) or the vernier. These instruments have their own scales, which are once again subject to parallax error.

Likewise, if these instruments are analog, we will have ammeters (current meter), ohmmeters (electrical resistance meter), barometers (atmospheric pressure meter), as well as many others, each with scales sensitive to parallax error.

In them, a needle or pointer will move along the scale, whose tip will indicate the appropriate reading as long as we fix our eyes as close and frontal as possible.

Astronomy

In the world of astronomy, parallax has a very special function: to measure distances between celestial bodies, such as planets and stars.

If you place a finger in front of your nose and close and open one eye at a time, you will notice how the finger appears from one side to the other. But this effect diminishes if it is repeated with the finger as far away from the face as possible. The closer a body is, the greater its parallax; and therefore the error that can be made by observing it is greater.

This is the basis for the use of parallax to determine how far away celestial bodies are: comparing the relative positions of two stars from different angles.

Examples of parallax errors

Wall Clock

If we read the time by looking at the wall clock from the left or right, we will get an incorrect time for several minutes. To see the correct time, the hands of the clock must be seen from the front.

Thermometer

Inside the thermometer there is a rising or falling liquid whose meniscus equals a temperature scale. If we view this meniscus from above or below, we will get incorrect temperatures. The thermometer must be viewed from the front and as vertical as possible.

Speedometer

Whoever drives can know precisely what the speed of the car is. However, the copilot (on the right) will see that the needle points at a speed apparently higher than the actual speed.

Preparation of solutions

When a solution is to be gauged, it must be done with the eyes just in front of the gauging line, because otherwise a parallax error equal to that of the thermometer will be repeated.

Analog blood pressure meter

Very similar to the case of the clock, or that of a stopwatch, is that of the blood pressure meter, whose needle must be seen from the front or it can be concluded that the pressure is lower or higher than the real one.

Granataria balance

In the granatary scales you have to read the readings from the front, because the same parallax error that occurs in, for example, school rules would be committed. Reading from the front the true mass of the object in question is obtained.

What is parallax error and why does it matter? Jim Tyler explains what causes parallax error in your scope, why it matters, and how to avoid it!

The term ‘parallax error’ is used to describe movement of the viewed image through a scope relative to the reticle, caused by a change in the position of the user’s eye relative to the axis of the scope. Put simply, the position of your intended pellet point of impact (POI) moves away from the reticle aim point if your eye moves away from being directly behind the axis of the scope. Parallax error is thus a potential cause of inaccuracy.

Simple Equation

The distance by which the image of the intended POI can move away from the reticle aim point can be calculated using a very simple equation, which is the radius of the objective lens multiplied by the difference between the target distance and the scope focus (parallax) range, and divided by twice the focus range.

A 40mm objective lens scope parallax focused at 25 yards and with a target at 40 yards has a maximum potential shift of 20 x 15/50, which works out at 6mm. That means the pellet can land a maximum of 6mm – in any direction according to whether your eye is left, right, high or low – away from the intended POI, and it’s important to realise that this figure is the absolute maximum, to achieve which, your eye must be a long way from the centre of the scope, much further than you’ll normally achieve unless you make a special effort to look through the edge of the image.

Experimental evidence

Experiments suggest that unless you do make a deliberate effort to move your eye as far from the scope axis as possible, parallax POI shift, in practice, is likely to be nearer a quarter of the absolute maximum, or 1.5mm, which isn’t a lot unless you’re a paper-target shooter.

There are ways in which the POI error caused by image shift can be reduced, or even negated, but before looking at them, we’ll consider what exactly causes the image shift that can lead to parallax error.

Focus

All the light we see when viewing a target in daylight emanates from the sun, and has reflected from the surface of the target. The light from a single point on the target bounces off in many directions, some of it entering the entire surface of the objective (front) lens of a telescopic sight; refracting (passing through) each air/glass boundary causes the light to bend, and the objective when focusing a scope for parallax purposes is to get all of the light from that single point on the face of the target to converge at a single point, at the plane of the reticle. When this is achieved, the light from that single point or any single point on the target, for that matter, will always appear in exactly the same place relative to the reticle, and parallax error simply cannot occur, no matter how far from the axis of the scope you place your eye.

Focusing systems

Sounds good, but there’s a fly in the ointment, which is that it is not straightforward to get the light to converge exactly at the plane of the reticle; before looking at that, a quick look at the means of focusing the scope.

Some scopes have one of two mechanisms for focusing the objective lens; the first to appear was an adjustable collar surrounding the objective bell, which is connected to the objective lens carrier, and can be screwed in and out to move the objective lens, and hence the point at which the light from a single point converges, back and forth. Scopes with this mechanism are usually advertised as PA (parallax adjustable).

The other, more recent, focus method is a wheel mounted on the side of the turret, which moves a lens or lenses in a carrier inside the scope to move the point of convergence back and forth, and these are usually advertised as SF (side focus). Both mechanisms do the same job, and the only advantage of the SF is that it’s arguably easier to adjust while looking through the scope.

The fly in the ointment

Remember I mentioned a fly in the ointment? Well, it is not enough for the light from the target to converge at the reticle, it also has to pass through the lens of your eye and converge again at a single point on your retina, and herein lies the potential problem.

The scope focus mechanism that controls the point of convergence in your eye is the ocular focus, at the rear of the scope, and it is necessary because our eyes are not all exactly the same.

The usual advice for focusing the ocular lens is to point the scope at a plain area of sky, and turn it until the image of the reticle is at its sharpest. The problem with doing it that way is that your eye has autofocus, and your eye might force focus with the plain background, but when your eye focuses instead at the image of a target, the reticle might consequently not be at its sharpest – you probably won’t notice it because you’re concentrating on the image of the target – because the light is not converging on your retina, and the consequence can be parallax error.

The right way

Set the ocular focus as best you can using the method described above, and set a target out at your chosen (parallax) focus range.

With your scope at its maximum magnification (to reduce depth of field, which makes it easier to judge when it’s in focus), point the scope at the target and adjust the front focus until the image is at its sharpest; then, taking care not to move the scope, move your head and see whether the image moves. If it does, readjust the ocular focus and try again, repeating until the image does not move relative to the reticle. The scope is now properly focused.

Other means

Hunters and field-target shooters can adjust focus to the range of every shot, but HFT shooters are not allowed to alter scope settings once they have started a course, and like those of us who use scopes that do not have a focus mechanism, they can reduce the potential for parallax error by other means. The key is always to look through the same part of the scope, preferably the centre, although what’s important is consistency. This can be achieved through practising whilst concentrating on head position until it becomes second nature, although that’s easier said than done in the three positions used in HFT.

More help

There is something else that can help limit parallax error, and that is proper gun fit. Few airguns have stock combs the right height for scopes held in high, medium or, in some cases, even low mounts, so they aren’t much help in trying to achieve consistent eye placement. The ideal, although most expensive solution is to have a custom stock tailor-made to suit you, or if you have the necessary skills, to make your own. A tailor-made stock won’t just help to reduce parallax error, but will also help with cheek weld and generally make your hold more consistent.

Cheaper option

A less expensive alternative is to have the comb cut from the stock, then refitted with riser bars that allow you to set the height to suit you. This has the advantage that if you come to sell the rifle or stock, the buyer can adjust it to suit him or herself.

The cheapest option is only for the brave, and it entails planing the comb level, glueing on a block of wood and reshaping it. I did this to a couple of stocks with good results many years ago, but these days, I’m not so brave.

Conclusion

In theory, parallax error can be significant, but in practice and with a little care, it can be minimised.

Parallax error is the error that is most committed when readings are taken in physics. You can thus understand why it is important to avoid it at all cost. One must be aware of its existence at all time so that it can be avoided and as a result the true value of the reading is obtained.

The concept of parallax error is related to the term parallax.

Imagine that we have  in a room a pelican and a flamingo as shown in fig 1 below.

Fig 1

Now Garfield is moving about in the room and each time is is somewhere in the room, he looks at the two birds. At the point A he sees the flamingo on the left of the pelican whereas when he is at position C he would see the flamingo on the right of the pelican. Only when he would be at position B would he sees the two birds one behind the other.

He would discover that each time he is at a different position, he would find that the position of the flamingo relative to the pelican has changed.

You can also have this effect when you are in front of a clock. If you move from side to side you would find that the time that you can read from the clock is different. 

So we can the understand that the parallax is the change in the apparent position of an object when the position of the observer changes.

Now let us look at the concept of parallax error. If you have placed a pencil on a metre rule and you are reading its length then just like in fig 1 above and in fig 2 below you can place you eye everywhere you want.

Fig 2

As you can see from fig 2 above I have chosen three position at which you can place your eye.

Clearly at these three position we can have the following reading

Reading at A  = 6.2 cm

Reading at B  = 5.8 cm

Reading at C = 5.5 cm

What you you think would be the correct reading?

The correct reading is would be obtained when the eye is placed at B.

Fig 3

Now there is a line from the tip of eye to the of the pencil  that continues up to the scale. this line is called the line of sight and the mark at which the line intersect the scale is the length of the pencil. This line of sight must be be at right angle to the scale. This is shown above in fig

If the line of sight and the scale are not ar right angle to each other then a parallax error is committed.

Similarly with a measuring cylinder the line of sight from the eye to the bottom of the meniscus must be at right angle to the scale as shown in fig 4 below. In this case the line of sight is horizontal and the scale vertical.

Fig 4

So as you can see above each time you are taking a reading you must ensure that the line of sight is perpendicular to the scale.

A measuring equipment can give precise but not accurate measurements, accurate but not precise measurements or neither precise nor accurate measurements.

Accuracy is a measure of how close the results of an experiment agree with the true value.

• When there is high accuracy (accurate shots), there will be small systematic error
• The accuracy of a reading can be improved by repeating the measurements.

Precision is a measure of how close the results of an experiment agree with each other. It is a measure of how reproducible the results are.

• High precision implies a small uncertainty and small random error.
• Precision is how close the measured values are to each other but they may not necessarily cluster about the true value.
• Zero errors and parallax errors affect the precision of an instrument.

Parallax Error

For accurate measurement, the eye must always be placed vertically above the mark being read. This is to avoid parallax errors which will give rise to inaccurate measurement.

Parallax errors affects the accuracy of the measurement. If you consistently used the incorrect angle to view the markings, your measurements will be displaced from the true values by the same amount. This is called systematic error.

However, if you used different angles to view the markings, your measurements will be displaced from the true values by different amounts. This is called random error.

Parallax error for micrometer screw gauge:

Zero Error

Zero Errors of Vernier Caliper

When the jaws are closed, the vernier zero mark coincides with the zero mark on its fixed main scale.

Before taking any reading it is good practice to close the jaws or faces of the instrument to make sure that the reading is zero. If it is not, then note the reading. This reading is called “zero error”.

The zero error is of two types:

1. Positive zero error; and
2. Negative zero error.

Positive Zero Error (Vernier calipers)

If the zero on the vernier scale is to the right of the main scale, then the error is said to be positive zero error and so the zero correction should be subtracted from the reading which is measured.

Negative Zero Error (Vernier calipers)

If the zero on the vernier scale is to the left of the main scale, then the error is said to be negative zero error and so the zero correction should be added from the reading which is measured.

Please refer to “How To Read A Vernier Caliper” for more information.

More Vernier Caliper Practice:

• Without Zero Error
• Finding The Zero Error
• With Zero Error
• With Other Measurement Topics

Zero Errors for Micrometer Screw Gauge

Positive Zero Error (Micrometer Screw Gauge)

If the zero marking on the thimble is below the datum line, the micrometer has a positive zero error. Whatever reading we take on this micrometer we would have to subtract the zero correction from the readings.

Negative Zero Error (Micrometer Screw Gauge)

If the zero marking on the thimble is above the datum line, the micrometer has a negative zero error. Whatever readings we take on this micrometer we would have to add the zero correction from the readings.

Note: You do not have to memorise positive error = subtract, negative error = add, just think this through for a while. It is rather straightforward and intuitive.

Please refer to “How To Read A Micrometer Screw Gauge” for more information.

Self-Test Questions

How can you avoid parallax errors when measuring a length with a metre rule?